相似和可能相交的族
Compressions and Probably Intersecting Families
摘要：
一个集合族X称为相交的，如果X的任何两个成员有非空的交集。这是一个众所周知的简单事实，即[n]={1，2，...，n}最多可以包含2^（n-1）个集合。卡托纳、卡托纳和卡托纳提出以下问题。相反，假设[n]的子集族X满足|X| =2^（n-1）+i，其中某个固定i>0。通过以某个固定的概率p独立地选择X的每个成员来创建一个新的家族X_p。我们如何选择X以最大化X_p相交的概率？他们猜想，对于i=1，2，.，2^（n-1）.本文证明了族[n]^（\ge r）={A \subset [n]：|A| \ge r}对于适当的i值是最优的，从而证明了关于这个值序列的猜想.此外，我们证明了，对于中间值的i，存在最佳的家庭之间躺在我们已经找到的。事实证明，我们找到的最佳家庭同时最大化的数量相交的子家庭的每一个可能的顺序。标准的压缩技术似乎不足以解决这个问题，因为它们不保留子族的相交属性。相反，我们的主要工具是一种新的压缩方法，以及一种“压缩”子家族的方式，这可能是独立的利益。
英文摘要：
A family X of sets is said to be intersecting if any two members of X have non-empty intersection. It is a well-known and simple fact that an intersecting family of subsets of [n]={1,2,...,n} can contain at most 2^(n-1) sets. Katona, Katona and Katona ask the following question. Suppose instead a family X of subsets of [n] satisfies |X|=2^(n-1)+i for some fixed i>0. Create a new family X_p by choosing each member of X independently with some fixed probability p. How do we choose X to maximize the probability that X_p is intersecting? They conjecture that there is a nested sequence of optimal families for i=1, 2, ..., 2^(n-1). In this paper, we show that the families [n]^(\ge r)={A\subset[n]:|A|\ge r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of every possible order. Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of `compressing' subfamilies, which may be of independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
